A hot spot spreads out and smooths away. One law governs it — ∂u/∂t = α ∂²u/∂x² — and it is runnable: a point of heat diffuses into a Gaussian whose width grows exactly like √t, the total heat is conserved, and the peak only ever falls. Down the center, data flows: a temperature profile goes in, the diffusion engine steps it forward, the spread comes out. The blue team builds and defends it; the red team tries to break it.
source Fourier, Théorie analytique de la chaleur (1822), F. Didot, Paris — archive.org/details/bub_gb_TDQJAAAAIAAJ. Rendered, not quoted.
The law is a balance: temperature changes in time at the rate its profile curves in space. ∂u/∂t = α ∂²u/∂x², with diffusivity α > 0 (units where length and time are natural, so α carries them).
A point source has a closed-form answer — the fundamental solution, a normalized Gaussian:
u(x,t) = 1/√(4παt) · exp(−x²/(4αt)) — width σ = √(2αt), spread √(4αt).
Live, measured off the running profile:
Fourier, 1822: heat obeys a parabolic PDE, and its solution operator is convolution with a spreading Gaussian — the heat kernel.
That same kernel is the transition density of the-brownian-motion: a random walker’s position after time t is Gaussian with variance 2αt. And convolving an image with it is exactly the-gaussian-blur. Diffusion, random walk, blur — one kernel, three doors. Each sphere is the next one’s premise.
The blue team’s live check: re-run the diffusion engine and confirm the invariants — normalized Gaussian, conserved total, non-increasing peak. If red tampers, this badge is where it shows.
The input is a starting temperature over a 1-D bar: u(x, 0). Pick a shape — a sharp point of heat, a bar held hot in the middle (a top-hat), two spots, or seeded lumps. The bar is insulated at both ends (no heat leaves), so with no source the total heat is fixed from the start.
This profile is what you feed the panel below. The engine never adds or removes heat — it only moves it, from where the curve is convex to where it is concave.
Forward heat flow: sharp features smooth, the peak falls, the area stays. Well-posed.
Drag time or press diffuse — every frame is recomputed from the four-rule stencil on the spot, never looked up.
What the machine produces, proven: a point source becomes a Gaussian whose width grows as √t (double the time → √2 the width), the total heat stays constant to floating-point, and the peak only decreases — the maximum principle. Left alone forever, the insulated bar reaches its steady state: ∂²u/∂x² = 0, a flat (harmonic) profile.
The blue team’s witness (left) confirms these live; the red team (right) tries to make them wrong.
It is also irreversible: run it backward and it is ill-posed — the tiniest noise explodes (that is the tamper, right). And it is linear: it ignores temperature-dependent conductivity, radiation (~T⁴), convection, and phase change. It is the first, cleanest law of spreading — not the last word on heat.
“Heat spreads at a finite speed here.” Cut. The fundamental solution is nonzero everywhere for any t>0 — infinite speed. The finite-speed fix is the hyperbolic Cattaneo/telegraph correction, a different equation.
“You can run it backward to recover the past.” Cut. The backward heat equation is ill-posed — exactly the tamper below.
“Total heat is only approximately conserved.” Corrected. With no source and insulated ends it is conserved exactly — the flux form makes Σ(∇²u) = 0 to floating point.
The red team’s move: flip the sign of α to negative — the backward heat equation. Now a smooth profile sharpens and blows up instead of smoothing. The maximum principle dies; the witness (window 7) is watching.
Set α < 0 and the engine runs time in reverse: high frequencies amplify, the peak grows, the witness recomputes, disagrees, and turns red. Nothing is faked; the attack is real and it is caught.